A sharp, distribution free, non-asymptotic result is proved for the concentration of a random function around the mean function, when the randomization is generated by a finite sequence of independent data and the random functions satisfy uniform bounded variation assumptions. The specific motivation for the work comes from the need for inference on the distributional impacts of social policy intervention. However, the family of randomized functions that we study is broad enough to cover wide-ranging applications. For example, we provide a Kolmogorov-Smirnov like test for randomized functions that are almost surely Lipschitz continuous, and novel tools for inference with heterogeneous treatment effects. A Dvoretzky-Kiefer-Wolfowitz like inequality is also provided for the sum of almost surely monotone random functions, extending the famous non-asymptotic work of Massart for empirical cumulative distribution functions generated by i.i.d. data, to settings without micro-clusters proposed by Canay, Santos, and Shaikh. We illustrate the relevance of our theoretical results for applied work via empirical applications. Notably, the proof of our main concentration result relies on a novel stochastic rendition of the fundamental result of Debreu, generally dubbed the "gap lemma," that transforms discontinuous utility representations of preorders into continuous utility representations, and on an envelope theorem of an infinite dimensional optimisation problem that we carefully construct.

翻译：本文针对随机函数在均值函数周围的集中性，证明了在随机化由有限独立数据序列生成且随机函数满足一致有界变差假设的条件下，一个精确、分布自由、非渐近的结果。该工作的具体动机源于对社会政策干预分布效应进行推断的需求。然而，我们所研究的随机函数族具有足够广泛的适用性，可覆盖多种应用场景。例如：为几乎必然 Lipschitz 连续的随机函数提供了类 Kolmogorov-Smirnov 检验，并为异质性处理效应的推断提供了新工具。此外，针对几乎必然单调随机函数之和，我们给出了类似 Dvoretzky-Kiefer-Wolfowitz 的不等式，将 Massart 关于独立同分布数据生成的经验累积分布函数的著名非渐近结果，推广至不存在 Canay、Santos 和 Shaikh 所提出的微观聚类情形。我们通过实证应用说明了理论结果对实际工作的意义。值得注意的是，我们的主要集中性结果的证明依赖于 Debreu 基本结果（通常称为"间隙引理"）的一个新颖随机化版本，该引理将预序的不连续效用表示转化为连续效用表示，并依赖于我们精心构造的无穷维优化问题的包络定理。